# Adiabatic evolution and floquet theorem

1. Oct 29, 2014

### paolorossi

Hello everybody,
I'm trying to understand if is possible to say something about the Floquet exponents, in the limit of a very slow changing on time. I try to explain. Given the differential equation
$$\dot{\vec{v}}(t) = A(t) \vec{v}(t)$$
with
$$A(t+T)=A(t)$$
a monodromy matrix is given by
$$\Phi(T)$$
where
$$\Phi(t)$$
is a matrix that is solution of
$$\dot{\Phi}(t) = A(t) \Phi(t)$$
and the Floquet exponents can be calculated from the spectrum of this monodromy matrix.
My question is:
can we calculate an expression for the Floquet exponents in terms of the eigenvalues of $$A(t)$$, in the limit $$T\rightarrow \infty$$ ?

2. Nov 3, 2014

### Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 4, 2014

### paolorossi

Thinking in terms of the quantum adiabatic theorem (QAT), maybe it's possible to do this under some restriction on the eigenvalues of the matrix A(t).

Anyway it's known that the solutions, i.e. the elements of the matrix \Phi, are bounded in time if the eigenvalues of A(t) are always imaginary. If I correctly remember, it's true that, if the eigenvalues of A(t) never cross the imaginary axis, then, when there is almost an eigenvalue of A(t) that has real part positive, the solutions are unbounded in time, i.e. there is almost a Floquet exponent with real part positive.

But, for example, I don't know if it's possible to find an expression of this Floquet exponent in terms of the eigenvalues of A(t). Thinking in terms of QAT, maybe in this limit the Floquet exponents play the role of the Berry phases, but I'm not sure.

I hope that someone that work with this kind of things can give me an help :)